MEN OF MATHEMATICS complex integers (numbers of the form a — bi , where a, 6 are rational integers and i denotes V — 1) into the higher arith- metic in order to give the law of biquadratic reciprocity its simplest expression. Dirichlet and other followers of Gauss then discussed quadratic forms in which- the rational integers appearing as variables and coefficients are replaced by Gaussian complex integers. Herrnite passed to the general case of this situation and investigated the representation of integers in what are to-day called Hermitian forms. An example of such a form (for the special case of two complex variables as^ x* and their 'conjugates* x^ x» instead of n variables) is + fl22 in 'which the bar over a letter denoting a complex number indi- cates the conjugate of that number; namely, if a? -f- iy is the complex number, its 'conjugate' is x — iy; and the coefficients <*ii» «i2t asi> fl22 are such that a% = ayt, for (ij) = (1,1), (1,2), (2,1), (2,2), so that a12 and azl are conjugates, and each of <zu, aZ2 is its own conjugate (so that an, a22 are real numbers). It is easily seen that the entire form is real (free of i) if all products are multiplied out, but it is most 'naturally' discussed in the shape given. \Vhen Hermite invented such forms he was interested in finding what numbers are represented by the forms. Over seventy years later it was found that the algebra of Hermitian forms is indispensable in mathematical physics, particularly in the modern quantum theory. Hermite had no idea that his pure mathematics would prove valuable in science long after his death. - indeed, like Archimedes, he never seemed to care much for the scientific applications of mathematics. But the fact that Hermite's work has given physics a useful tool is perhaps another argument favouring the side that believes mathemati- cians best justify their abstract existence when left to their own inscrutable devices. Leaving aside Hennite's splendid discoveries in the theory of algebraic invariants as too technical for discussion here, we shall pass on in a moment to two of his most spectacular achieve- ments in other fields. The high esteem in which Hennite's 506